Asked by he
Line AB
is on the coordinate plane and does not pass through the origin. Line AB
is dilated with a scale factor of 3 and a center of dilation at the origin to create line A′B′
. Describe the effects of the dilation on Line AB
. If line AB
was dilated with a scale factor of 1/3
, how would your answer change?(4 points)
All Answers 1
Answered by
GPT-5 mini
AI
Let the dilation center be the origin and the scale factor be k. Any point (x,y) on line AB maps to (kx, ky) on A′B′.
For k = 3 (scale factor 3):
1. A′B′ is parallel to AB (slope is unchanged).
2. A′B′ does not pass through the origin (since AB did not).
3. Every point on AB is moved three times as far from the origin, so the perpendicular distance from the origin to A′B′ is 3 times the distance to AB.
4. If AB has equation ax + by + c = 0 (c ≠ 0), then A′B′ has equation ax + by + 3c = 0 (coefficients a and b same, c multiplied by 3).
If the scale factor were 1/3 instead:
- All four statements remain true with “3” replaced by “1/3”: A′B′ is parallel to AB, does not pass through the origin, its distance from the origin is one-third the original distance, and its equation would be ax + by + (1/3)c = 0.
For k = 3 (scale factor 3):
1. A′B′ is parallel to AB (slope is unchanged).
2. A′B′ does not pass through the origin (since AB did not).
3. Every point on AB is moved three times as far from the origin, so the perpendicular distance from the origin to A′B′ is 3 times the distance to AB.
4. If AB has equation ax + by + c = 0 (c ≠ 0), then A′B′ has equation ax + by + 3c = 0 (coefficients a and b same, c multiplied by 3).
If the scale factor were 1/3 instead:
- All four statements remain true with “3” replaced by “1/3”: A′B′ is parallel to AB, does not pass through the origin, its distance from the origin is one-third the original distance, and its equation would be ax + by + (1/3)c = 0.
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